<ul><li>This post focuses on the range 3 ≤ r ≤ 1 + √6 of the logistic map, analyzing the first fork of the image where it splits into two stable branches.</li><li>For r values between 3 and 1 + √6, there is no stable fixed point but a stable cycle is present, with points near one cycle point transitioning to the other.</li><li>Iterations starting with x = 0.1 and r = π demonstrate convergence to two cycle points at 0.5373 and 0.7810, with the first 20 iterates provided.</li><li>As r exceeds 1 + √6, stable cycle points become unstable, leading to period doubling where new cycle points emerge, eventually reaching chaotic behavior beyond r = 3.56995.</li></ul>

The first fork

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